Lower Bounds for Oblivious Subspace Embeddings

An oblivious subspace embedding (OSE) for some epsilon, delta is an element of (0, 1/3) and d <= m <= n is a distribution D over R-mxn such that P-Pi similar to D (for all x is an element of W, (1 - epsilon) parallel to x parallel to 2 <= parallel to Pi x parallel to 2 = (1 + epsilon)parallel to x parallel to 2) >= 1 - delta for any linear subspace W subset of R-n of dimension d. We prove any OSE with delta < 1/3 has m = Omega((d + log(1/delta))/epsilon(2)), which is optimal. Furthermore, if every. in the support of D is sparse, having at most s non-zero entries per column, we show tradeoff lower bounds between m and s.